Dirac Operators in Riemannian Geometry (Graduate Studies in Mathematics)

For a Riemannian manifold $M$, the geometry, topology and analysis are interrelated in ways that are widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin (or $\textrm{spin}^\mathbb{C}$) structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten invariants. In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\textrm{spin}^\mathbb{C}$ structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on $M$ lead to results about whether $M$ is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections. This detailed book with elegant proofs is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas. This edition is translated from the German edition published by Vieweg Verlag. Front Cover......Page 1 Title......Page 4 Copyright......Page 5 Contents......Page 6 Introduction......Page 10 1.1. Linear algebra of quadratic forms ......Page 16 1.2. The Clifford algebra of a quadratic form ......Page 19 1.3. Clifford algebras of real negative definite quadratic forms ......Page 25 1.4. The pin and the spin group ......Page 29 1.5. The spin representation ......Page 35 1.6. The group Spin ......Page 40 1.7. Real and quaternionic structures in the space of n-spinors ......Page 44 1.8. References and exercises ......Page 47 2.1. Spin structures on SO(n)-principal bundles ......Page 50 2.2. Spin structures in covering spaces ......Page 57 2.3. Spin structures on G-principal bundles ......Page 60 2.4. Existence of spin structures ......Page 62 2.5. Associated spinor bundles ......Page 68 2.6. References and exercises ......Page 71 3.1. Connections in spinor bundles ......Page 72 3.2. The Dirac and the Laplace operator in the spinor bundle ......Page 82 3.3. The Schrodinger-Lichnerowicz formula ......Page 86 3.4. Hermitian manifolds and spinors ......Page 88 3.5. The Dirac operator of a Riemannian symmetric space ......Page 97 3.6. References and Exercises ......Page 103 4.1. The essential self-adjointness of the Dirac operator in L^2 ......Page 106 4.2. The spectrum of Dirac operators over compact manifolds ......Page 113 4.3. Dirac operators are Fredholm operators ......Page 122 4.4. References and Exercises ......Page 126 5.1. Lower estimates for the eigenvalues of the Dirac operator ......Page 128 5.2. Riemannian manifolds with Killing spinors ......Page 131 5.3. The twistor equation ......Page 136 5.4. Upper estimates for the eigenvalues of the Dirac operator ......Page 140 5.5. References and Exercises ......Page 142 A.1. On the topology of 4-dimensional manifolds ......Page 144 A.2. The Seiberg-Witten equation ......Page 149 A.3. The Seiberg-Witten invariant ......Page 153 A.4. Vanishing theorems ......Page 159 A.S. The case dim ML (g) = 0 ......Page 161 A.6. The Kahler case ......Page 162 A.7. References ......Page 168 B.1. Principal fibre bundles ......Page 170 B.2. The classification of principal bundles ......Page 177 B.3. Connections in principal bundles ......Page 178 B.4. Absolute differential and curvature ......Page 181 B.5. Connections in U(1)-principal bundles and the Weyl theorem ......Page 184 B.6. Reductions of connections ......Page 188 B.7. Frobenius' theorem ......Page 189 B.9. Holonomy theory ......Page 192 B.10. References ......Page 193 Bibliography ......Page 194 Index ......Page 208 Back Cover......Page 211 Front Cover 1 Title 4 Copyright 5 Contents 6 Introduction 10 Chapter 1. Clifford Algebras and Spin Representation 16 1.1. Linear algebra of quadratic forms 16 1.2. The Clifford algebra of a quadratic form 19 1.3. Clifford algebras of real negative definite quadratic forms 25 1.4. The pin and the spin group 29 1.5. The spin representation 35 1.6. The group Spin 40 1.7. Real and quaternionic structures in the space of n-spinors 44 1.8. References and exercises 47 Chapter 2. Spin Structures 50 2.1. Spin structures on SO(n)-principal bundles 50 2.2. Spin structures in covering spaces 57 2.3. Spin structures on G-principal bundles 60 2.4. Existence of spin structures 62 2.5. Associated spinor bundles 68 2.6. References and exercises 71 Chapter 3. Dirac Operators 72 3.1. Connections in spinor bundles 72 3.2. The Dirac and the Laplace operator in the spinor bundle 82 3.3. The Schrodinger-Lichnerowicz formula 86 3.4. Hermitian manifolds and spinors 88 3.5. The Dirac operator of a Riemannian symmetric space 97 3.6. References and Exercises 103 Chapter 4. Analytical Properties of Dirac Operators 106 4.1. The essential self-adjointness of the Dirac operator in L^2 106 4.2. The spectrum of Dirac operators over compact manifolds 113 4.3. Dirac operators are Fredholm operators 122 4.4. References and Exercises 126 Chapter 5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors 128 5.1. Lower estimates for the eigenvalues of the Dirac operator 128 5.2. Riemannian manifolds with Killing spinors 131 5.3. The twistor equation 136 5.4. Upper estimates for the eigenvalues of the Dirac operator 140 5.5. References and Exercises 142 Appendix A. Seiberg-Witten Invariants 144 A.1. On the topology of 4-dimensional manifolds 144 A.2. The Seiberg-Witten equation 149 A.3. The Seiberg-Witten invariant 153 A.4. Vanishing theorems 159 A.S. The case dim ML (g) = 0 161 A.6. The Kahler case 162 A.7. References 168 Appendix B. Principal Bundles and Connections 170 B.1. Principal fibre bundles 170 B.2. The classification of principal bundles 177 B.3. Connections in principal bundles 178 B.4. Absolute differential and curvature 181 B.5. Connections in U(1)-principal bundles and the Weyl theorem 184 B.6. Reductions of connections 188 B.7. Frobenius' theorem 189 B.8. The Freudenthal-Yamabe theorem 192 B.9. Holonomy theory 192 B.10. References 193 Bibliography 194 Index 208 Back Cover 211 For a Riemannian manifold $M$, the geometry, topology and analysis are interrelated in ways that are widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin (or $\mathrm{spin}^\mathbb{C}$) structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten invariants. In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\mathrm{spin}^\mathbb{C}$ structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on $M$ lead to results about whether $M$ is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections. This detailed book with elegant proofs is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas. This edition is translated from the German edition published by Vieweg Verlag. Chapter 1. Clifford Algebras And Spin Representation Chapter 2. Spin Structures Chapter 3. Dirac Operators Chapter 4. Analytical Properties Of Dirac Operators Chapter 5. Eigenvalue Estimates For The Dirac Operator And Twistor Spinors Appendix A. Seiberg-witten Invariants Appendix B. Principal Bundles And Connections Thomas Friedrich ; Translated By Andreas Nestke. Includes Bibliographical References And Index.